A model is a mathematical relationship that comes with a story. Stokey and Zeckhauser (1978) give a definition: “A model is a simplified representation of some aspect of the real world, sometimes of an object, sometimes of a situation or a process”.
A good model reduces a complex situation to a set of essential mechanisms, or dynamics, that an analyst needs in order to make a good decision.
A bad model mis-characterizes the mechanism of interest, is too simple to capture important dynamics, or is too complicated to be calibrated or understood.
More: Otto and Day (2011), Frigg and Hartmann (2012), Basu and Andrews (2013), Heesterbeek et al. (2015)
All models are wrong, but some are useful. – George Box
A good model is suited to a particular problem, and balances parsimony and realism, simplicity and complexity.
Sometimes the corresponding empirical study may be infeasible or unethical to conduct in real life.
For example,
What would happen if every injection drug user had access to naloxone? How many fatal overdoses would be averted? Is this intervention program cost-effective?
What would happen if the government eliminated funding for smoking cessation programs?
Models formalize scientific hypotheses about the mechanism that produces a phenomenon of interest.
When data agree with our model, then we may accumulate evidence that the hypotheses underlying the model are reasonable.
When we observe data that do not agree with the predictions of our model, then this might be evidence that our hypotheses are wrong.
Krebs et al. (2019)
Crawford, Wu, and Heimer (2018)
Pitzer et al. (2012)
Bilcke et al. (2019)
Wein, Craft, and Kaplan (2003)
Kaplan, Craft, and Wein (2002)
Kaplan (1995)
Gonsalves, Kaplan, and Paltiel (2015)
Stokey and Zeckhauser (1978), Vynnycky and White (2010), Otto and Day (2011)
Question: How many people inject drugs (e.g. opioids) in my city?
Data: counts of \(m\) individuals’ emergency room visits for overdose, \(X_1,\ldots,X_m\), all positive, for one unit of time (e.g. year). We only see \(X_i\) if person \(i\) had at least one overdose.
Why is this a hard problem? It seems that we do not have enough data to learn what we want to know!
Let’s illustrate a modeling-based solution using a stylized depiction of the overdoses. This is a preview of the mindset and recipe for modeling that you will learn about in this course.
Let \(N\) be the number of people who inject drugs in the city.
Let \(X_1,\ldots,X_N\) be the number of times (possibly zero) each has overdosed and been taken to the emergency room, in one year.
Let \(M=m\) the number who have had at least one overdose. We know \(X_1,\ldots,X_m > 0\)
Assume everyone who has had at least one overdose went to the emergency room.
Assume every drug injector has an overdose with constant rate \(\lambda\) per year. (This is a gross simplification of reality!)
We can use the constant rate assumption to learn about \(\lambda\) from \(X_1,\ldots,X_m\).
(In fact, \(X_i\sim\text{Poisson}(\lambda)\), so we can estimate \(\lambda\) from the data)
Then, we know that \[ M \approx \Pr(X_i>0) \times (\text{number at risk}) \] and it turns out that \[ M \approx (1-e^{-\lambda}) N . \]
Rearranging, we have the estimate
\[ \hat{N} = \frac{m}{1-e^{-\hat\lambda}} \]
Where was the magic step?
Specifying a common rate of overdose per drug user
This allows estimation of \(\lambda\), and implies a probability distribution for \(M\), which we can use to estimate \(N\).
Some questions:
Models are useful when the are:
Models are dangerous when they:
If you have taken a statistics class, you have seen statistical approaches to explaining variation. For example, consider the “statistical regression model” \[ y = \alpha + \beta x + \epsilon \] If we regard \(x\) as a treatment and \(y\) as a health outcome for a given patient, then we would like to think of \(\beta\) as the “effect” of the treatment.
This model posits a linear relationship between treatment and outcome. Given a one-unit change in \(x\), we expect the outcome \(y\) to change by an increment of \(\beta\).
We think there is no difference between “statistical” and “mechanistic” models, except for the stories we tell about their structure and coefficients. I think:
Basu, Sanjay, and Jason Andrews. 2013. “Complexity in Mathematical Models of Public Health Policies: A Guide for Consumers of Models.” PLoS Medicine 10 (10). Public Library of Science: e1001540.
Bilcke, Joke, Marina Antillón, Zoë Pieters, Elise Kuylen, Linda Abboud, Kathleen M Neuzil, Andrew J Pollard, A David Paltiel, and Virginia E Pitzer. 2019. “Cost-Effectiveness of Routine and Campaign Use of Typhoid Vi-Conjugate Vaccine in Gavi-Eligible Countries: A Modelling Study.” The Lancet Infectious Diseases. Elsevier.
Frigg, Roman, and Stephan Hartmann. 2012. “Models in Science, Stanford Encyclopedia of Philosophy.” https://plato.stanford.edu/entries/models-science/.
Gonsalves, Gregg S, Edward H Kaplan, and A David Paltiel. 2015. “Reducing Sexual Violence by Increasing the Supply of Toilets in Khayelitsha, South Africa: A Mathematical Model.” PLoS One 10 (4). Public Library of Science: e0122244.
Heesterbeek, Hans, Roy M Anderson, Viggo Andreasen, Shweta Bansal, Daniela De Angelis, Chris Dye, Ken TD Eames, et al. 2015. “Modeling Infectious Disease Dynamics in the Complex Landscape of Global Health.” Science 347 (6227): aaa4339.
Kaplan, Edward H. 1995. “Probability Models of Needle Exchange.” Operations Research 43 (4). INFORMS: 558–69.
Kaplan, Edward H, David L Craft, and Lawrence M Wein. 2002. “Emergency Response to a Smallpox Attack: The Case for Mass Vaccination.” Proceedings of the National Academy of Sciences 99 (16). National Acad Sciences: 10935–40.
Krebs, Emanuel, Benjamin Enns, Linwei Wang, Xiao Zang, Dimitra Panagiotoglou, Carlos Del Rio, Julia Dombrowski, et al. 2019. “Developing a Dynamic Hiv Transmission Model for 6 Us Cities: An Evidence Synthesis.” PloS One 14 (5). Public Library of Science: e0217559.
Otto, Sarah P, and Troy Day. 2011. A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution. Princeton University Press.
Pitzer, Virginia E, Katherine E Atkins, Birgitte Freiesleben de Blasio, Thierry Van Effelterre, Christina J Atchison, John P Harris, Eunha Shim, et al. 2012. “Direct and Indirect Effects of Rotavirus Vaccination: Comparing Predictions from Transmission Dynamic Models.” PloS One 7 (8). Public Library of Science: e42320.
Stokey, Edith, and Richard Zeckhauser. 1978. Primer for Policy Analysis. WW Norton.
Vynnycky, Emilia, and Richard White. 2010. An Introduction to Infectious Disease Modelling. Oxford University Press.
Wein, Lawrence M, David L Craft, and Edward H Kaplan. 2003. “Emergency Response to an Anthrax Attack.” Proceedings of the National Academy of Sciences 100 (7). National Acad Sciences: 4346–51.